This particular thermostat, a Honeywell, had no fewer than five settings, including one at the top of the list for "gas," and another, further down the list for "gas, more than 90% efficient." It was set for "gas, standard efficiency."
So back to Google we went, to discover that the furnace in question, a Trane XE 90, is indeed more than 90% efficient. (If you can easily see your furnace, a white plastic exhaust pipe or chimney is a sure sign of this.) We changed the setting, and presto: the furnace no longer cycles on and off constantly, and the house seems more comfortable and less drafty.
While you're at it, change the furnace filter, too; you'll get cleaner air in the house and help increase efficiency a bit because the air can flow more easily.